Abstract

Special arithmetic series \(f(x)=\sum_{n=0}^\infty c_nx^n\), whose coefficients \(c_n\) are normally given as certain binomial sums,
satisfy `self-replicating' functional identities. For example, the equation
$$
\frac1{(1+4z)^2}\,f\biggl(\frac z{(1+4z)^3}\biggr)
=\frac1{(1+2z)^2}\,f\biggl(\frac{z^2}{(1+2z)^3}\biggr)
$$
generates a modular form \(f(x)\) of weight 2 and level 7, when a related modular parametrization \(x=x(\tau)\) is properly chosen.
In this note we investigate the potential
of describing modular forms by such self-replicating equations as well as applications
of the equations that do not make use of the modularity. In particular, we outline a new recipe of generating AGM-type algorithms
for computing \(\pi\) and other related constants. Finally, we indicate some possibilities to extend the functional equations
to a two-variable setting.